Circles

1103077205

Level: 
B
A farmer has a fenced rhombus shaped garden with a side length of \( 4\,\mathrm{m} \). At one corner where the angle between the sides is \( 60^{\circ} \) the farmer tied a goat (see the picture). Of what length has to be the rope so that the goat grazes down exactly half the area of the garden? Round the result to one decimal place.
\( 3.6\,\mathrm{m} \)
\( 3.2\,\mathrm{m} \)
\( 4.1\,\mathrm{m} \)
\( 2.9\,\mathrm{m} \)

1103077209

Level: 
B
A semicircle is inscribed in a triangle \( KLM \) so that the diameter of the semicircle is parallel to side \( KL \) (see the picture). The length of \( KL \) is \( 8\,\mathrm{cm} \) and the height to side \( KL \) is \( 4\,\mathrm{cm} \) long. Determine the radius of the semicircle.
\( 2\,\mathrm{cm} \)
\( 4\,\mathrm{cm} \)
\( 6\,\mathrm{cm} \)
\( 8\,\mathrm{cm} \)

1103077210

Level: 
B
The picture shows a roundabout with the radius of \( 6\,\mathrm{m} \). Inside the roundabout there is a flower bed, which has the shape of an equilateral triangle inscribed in it. The remaining part inside the roundabout is the lawn. Calculate the area of the lawn.
\( 66.33\,\mathrm{cm}^2 \)
\( 46.77\,\mathrm{cm}^2 \)
\( 113.10\,\mathrm{cm}^2 \)
\( 24.66\,\mathrm{cm}^2 \)

1103021602

Level: 
C
The side of an equilateral triangle is \( 6\,\mathrm{cm} \) long. Find the area of the annulus between the incircle and circumcircle of the given triangle. (See the picture.)
\( 9\pi\,\mathrm{cm}^2 \)
\( 6\pi\,\mathrm{cm}^2 \)
\( 12\pi\,\mathrm{cm}^2 \)
\( 8\pi\,\mathrm{cm}^2 \)

1103021612

Level: 
C
Consider two circles: the circle \( k \) with centre \( S_1 \) and radius \( 3\,\mathrm{cm} \), and the circle \( n \) with centre \( S_2 \) and radius \( 8\,\mathrm{cm} \). The distance between \( S_1 \) and \( S_2 \) is \( 22\,\mathrm{cm} \). Common internal tangents of the circles intersect at point \( A \). Calculate the distance of the point \( A \) from the centre \( S_1 \). (See the picture.)
\( 6\,\mathrm{cm} \)
\( 16\,\mathrm{cm} \)
\( 11\,\mathrm{cm} \)
\( 5\,\mathrm{cm} \)

1103077106

Level: 
C
Let an equilateral triangle have a side of length of \( 10\,\mathrm{cm} \). Suppose there is a circular sector inside the triangle that has the centre at one of the vertices of the triangle, and the arc touches the opposite side (see the picture). Calculate the length of the arc of the sector. Round the result to two decimal places.
\( 9.07\,\mathrm{cm} \)
\( 8.62\,\mathrm{cm} \)
\( 8.93\,\mathrm{cm} \)
\( 9.05\,\mathrm{cm} \)

1103077107

Level: 
C
The figure shows an equilateral triangle whose side is \( 10\,\mathrm{cm} \) long. The circular sector inside the triangle has the centre at one of the vertices of the triangle, and the arc touches the opposite side. Find the ratio of the circumference of the sector to the perimeter of the triangle. Round the result to one decimal place.
\( 0.9 \)
\( 0.5 \)
\( 0.8 \)
\( 1.5 \)

1103077108

Level: 
C
The figure shows an equilateral triangle whose side is \( 10\,\mathrm{cm} \) long. The circular sector inside the triangle has the centre at one of the vertices of the triangle, and the arc touches the opposite side. Calculate the area of the sector. Round the result to one decimal place.
\( 39.3\,\mathrm{cm}^2 \)
\( 37.5\,\mathrm{cm}^2 \)
\( 14.4\,\mathrm{cm}^2 \)
\( 3.75\,\mathrm{cm}^2 \)

1103077109

Level: 
C
Two quarter circles are inscribed in the square with sides of \( 2\,\mathrm{dm} \). The centres of the quarter circles are at the opposite vertices of the square (see the picture). Calculate the area of the region between the quarter circles. Round the result to two decimal places.
\( 2.28\,\mathrm{dm}^2 \)
\( 3.14\,\mathrm{dm}^2 \)
\( 21.12\,\mathrm{dm}^2 \)
\( 1.72\,\mathrm{dm}^2 \)